Is 10 closer to infinity than 1?

This may be considered a philosophical question but is the number "10" closer to infinity than the number "1"?

There is more than one notion of "closer than."

Metric: We are not able to extend the metric on $\Bbb R$ to the extended line $[-\infty,\infty]$ in a meaningful way (i.e. where bigger elements of $\Bbb R$ are closer to $+\infty$ than smaller elements). Even if we extend the concept of "metric" to include infinite distances, this doesn't work.

Order-theoretic: A partial order $\le$ on a set is a relation satisfying certain properties. It is a way of ordering the elements of the set, put simply. A linear ordering is one in which every two elements are comparable: for any $x$ and $y$, one of $x\le y$ or $y\le x$ is true at least. The extended real line gets its linear ordering from $\Bbb R$, with $+\infty$ made maximal and $-\infty$ made minimal. If $x,y\le z$ then we can say that $y$ is closer than $x$ to $z$ if $x\le y\le z$ and $x\ne y$. On this view, $10$ is indeed closer to $+\infty$ than $1$ (and we can reverse directions to say $1$ is closer to $-\infty$ than $10$).

One observation to be made here is that we essentially decreed larger numbers closer to $+\infty$ purely by fiat. Rather than making this whole discussion trivial, though, it actually illustrates the power of modern algebra in converting imagination into (mathematical) reality. It makes intuitive sense for bigger numbers to be closer to $+\infty$, and there is a consistent way to encode this mathematically, so we simply make it so and therefore have what we want.

Topological: A metric on a space induces a topology on that space. In particular, a metrizable topology. There are more general topologies though, so we can do and allow more things. However not everything can be compared in terms of "closer than." One possible interpretation is that $x$ is closer than $y$ to $z$ if neighborhoods of $z$ containing $y$ also contain $x$ but not vice-versa. It is generally too optimistic to expect elements to be comparable in this way, but the topology on $\Bbb R$ and the extended line is nice enough to make this an almost useful concept.

The problem with this approach is that neigborhoods can be "disconnected," so we can simply put a neighborhood around $+\infty$ containing both $1$ and $10$ and then delete a patch around $10$, thus telling us that neither $1$ nor $10$ is closer than the other. But if we strengthen our notion to only connected neighborhoods, then we get what we want: every connected neighborhood of $+\infty$ containing $1$ must also contain $10$, but not vice-versa. [This originally slipped my mind, but Mike helpfully pointed it out.]

It is worth noting that the topology on the extended line is induced by the ordering on it (topologies are induced by orderings by considering "intervals" to be a topological base). So this actually isn't fundamentally any newer information than the previous context. Furthermore, different metrics can induce the same topology, and it is possible to make $[-\infty,+\infty]$ a metric space in which $10$ is closer than $1$ to $+\infty$. For example, $d(a,b):=|\tan^{-1}(a)-\tan^{-1}(b)|$ does this, as Rahul points out

If you give a suitable precise meaning to "closer", then the answer can be yes.

Here are two things that true for the extended real number line, both of which could reasonably be informally described the way you suggest:

• $10$ lies between $1$ and $+\infty$.
• There are intervals containing $+\infty$ and $10$ that do not contain $1$. However, every interval containing $+\infty$ and $1$ also contains $10$

"Just follow that line forever," said the Mathemagician, "and when you reach the end, turn left. There you'll find the land of Infinity, where the tallest, the shortest, the biggest, the smallest, and the most and the least of everything are kept."

"I really don't have that much time," said Milo anxiously. "Isn't there a quicker way?"

"Well, you might try this flight of stairs," he suggested, opening another door and pointing up. "It goes there, too."

Milo bounded across the room and started up the stairs two at a time. "Wait for me, please," he shouted to Tock and the Humbug. "I'll be gone just a few minutes."

―Norton Juster, The Phantom Tollbooth